UNIT V JACOBIANS Dr T VENKATESAN Assistant Professor

UNIT V JACOBIANS Dr. T. VENKATESAN Assistant Professor Department of Statistics St. Joseph’s College, Trichy-2.

Lecture 5: Jacobians • In 1 D problems we are used to a simple change of variables, e. g. from x to u 1 D Jacobian maps strips of width dx to strips of width du • Example: Substitute

2 D Jacobian • For a continuous 1 -to-1 transformation from (x, y) to (u, v) • Then • Where Region (in the xy plane) maps onto region in the uv plane 2 D Jacobian maps areas dxdy to areas dudv • Hereafter call such terms etc

Why the 2 D Jacobian works • Transformation T yield distorted grid of lines of constant u and constant v • For small du and dv, rectangles map onto parallelograms • This is a Jacobian, i. e. the determinant of the Jacobian Matrix

Relation between Jacobians • The Jacobian matrix is the inverse matrix of i. e. , • Because (and similarly for dy) • This makes sense because Jacobians measure the relative areas of dxdy and dudv, i. e • So

Simple 2 D Example Area of circle A= r

Harder 2 D Example where R is this region of the xy plane, which maps to R’ here 8 4 1 9

An Important 2 D Example • Evaluate a • First consider a -a -a • Put • as

3 D Jacobian • maps volumes (consisting of small cubes of volume • . . . . to small cubes of volume • Where

3 D Example • Transformation of volume elements between Cartesian and spherical polar coordinate systems (see Lecture 4)